A note on a theorem of heath

We propose a generalization of Heath's theorem that semi-metric spaces with point-countable bases are developable: A semi-metrizable space X is developabale if (and only if) there is on it a σ-discrete family $\text{C}\text{=?}{}_{\mathrm{m?N}}\text{C}{}_{\mathrm{m}}$ of closed sets, interior-preserving over each member C of which is a countable family {$\text{D}$_n(C): n ∈ N} of collections of open sets such that if U is a neighbourhood of ξ∈X, then there are such a Γ∈$\text{C}$ and such a v∈ N that ξ ? Γ and ξ∈ int ∩ (D: ξ: D∈$\text{D}$_v(Γ))?U.     

The evaluation identification in function spaces

A study is made of the natural function which maps each point x of a space X to the evaluation function e_x:Y^x→Y defined by $\text{e}{}_{\mathrm{x}}\text{(?)=?(x)}$. A consequence of the results is that βX and υX can both be considered as subspaces of spaces of continous functions from appropriate domain spaces into I or R, respectively.     

A class of extension properties that are not simply generated

Let $\text{P}$ be a closed-hereditary topological property preserved by products. Call a space $\text{P}$-regular if it is homeomorphic to a subspace of a product of spaces with $\text{P}$. Suppose that each $\text{P}$-regular space possesses a $\text{P}$-regular compactification. It is well-known that each $\text{P}$-regular space X is densely embedded in a unique space γ_scPX with $\text{P}$ such that if f: X → Y is continuous and Y has $\text{P}$, then f extends continuously to γ_scPX. Call $\text{P}$-pseudocompact if γ_scPX is compact.Associated with $\text{P}$ is another topological property $\text{P}$^#, possessing all the properties hypothesized for $\text{P}$ above, defined as follows: a $\text{P}$-regular space X has $\text{P}$^# if each $\text{P}$-pseudocompact closed subspace of X is compact. It is known that the $\text{P}$-pseudocompact spaces coincide with the $\text{P}$^#-pseudocompact spaces, and that $\text{P}$^# is the largest closed-hereditary, productive property for which this is the case. In this paper we prove that if $\text{P}$ is not the property of being compact and $\text{P}$-regular, then $\text{P}$^# is not simply generated; in other words, there does not exist a space E such that the spaces with $\text{P}$^# are precisely those spaces homeomorphic to closed subspaces of powers of E.     

Spaces N ∪ R and their dimensions

About spaces N∪$\text{R}$ (see [2, Exercise 5I]), the following are proved: (1) dim $\text{N∪}\text{R}\text{= dim β(N∪}\text{R}\text{)?N∪}\text{R}\text{,(2)if|β(N∪}\text{R}\text{)?N∪}\text{R}\text{|<2}{}^{\mathrm{?}}{}^{\mathrm{o}}$, then no real-valued continuous fu ction on N∪$\text{R}$ is onto (and hence, dim $\text{N∪}\text{R}\text{=0}$), (3) any compact metric space without isolated points is homeomorphic to some $\text{β(N∪}\text{R}\text{)?N∪}\text{R}$ and (4)there are spaces X,X₁ and X₂ of the form N∪$\text{R}$ such that X=X₁∪X₂,X₂andX₂ are zero sets of X, and dim X=n, dimX₁=dimX₂=0, where n=1,2,… or ∞.     

Compactness in translation invariant Banach spaces of distributions and compact multipliers

It is shown that for a comprehensive family of translation invariant Banach spaces (B, ∥ ∥_B) of (classes of) measurable functions or distributions on a locally compact group (including most of the spaces of interest in harmonic analysis) the following compactness criterion generalizing the well-known results due to Kolmogorov-Riesz-Weil concerning compact sets in L^p(G), 1 ? p < ∞, holds true: A closed subset M ? B is compact in B if and only if it satisfies the following conditions: (a) sup_{? ? M} ∥?∥_B < ∞; (b) $\text{? ? > 0 ?k ∈ K(G):∥k1???∥}{}_{\mathrm{B}}\text{? ?}$ for all $\text{?∈M}$; (c) $\text{?? > 0 ?h∈K(G):∥h???∥}{}_{\mathrm{B}}\text{? ?}$ for all $\text{?∈M}$. Among various applications a characterization of the space of all compact multipliers between suitable pairs of such spaces can be derived.     

On projections and limit mappings of inverse systems of compact spaces

A compactificaton αX of a completely regular space X is “determined” by a subset F of C¹(X) if αX is the smallest compactificaton of X to which each element of F extends, and is “generated” by F if the evaluation map $\text{e}{}_{\mathrm{F}}\text{:X →}\text{R}{}^{\mathrm{n}}\text{,n = |F|}$, is an embedding and $\text{αX =}\text{e}{}_{\mathrm{F}}\text{(X)}$. Evidently, if F either determines or generates αX, then every elements of F has an extension to αX; whenever F satisfies this latter condition, the set of all such extensions is denoted F^α.A major results of our previous paper is that F determines αX if and only if F^α separates points of αX ? X. A major result of the present paper is that F generates αX if and only if F^α separates points of αX.     

The category of Urysohn spaces is not cowellpowered

It is shown that the category of Urysohn spaces and continuous maps is not cowellpowered. To this end we will construct for each ordinal number β a Urysohn space Y_β with card $\text{(Y}{}_{\mathrm{\beta }}\text{= ?}{}_0\text{?}\text{card}\text{(β)}$ and a continuous map $\text{e}{}_{\mathrm{\beta }}\text{:}\text{Q}\text{→ Y}{}_{\mathrm{\beta }}$ from the rationals into Y_β. It turns out that $\text{e}{}_{\mathrm{\beta }}$ is an external monomorphism in the category of Hausdorff spaces and an epimorphism in the category of Urysohn spaces.     

Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space

Let {X_n} be a ?-irreducible Markov chain on an arbitrary space. Sufficient conditions are given under which the chain is ergodic or recurrent. These extend known results for chains on a countable state space. In particular, it is shown that if the space is a normed topological space, then under some continuity conditions on the transition probabilities of {X_n} the conditions for ergodicity will be met if there is a compact set K and an ? > 0 such that $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 — 6X}{}_{\mathrm{n}}\text{6 ∣ X}{}_{\mathrm{n}}\text{= x} ? ??}$ whenever x lies outside K and $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 ∣ X}{}_{\mathrm{n}}\text{=x}}$ is bounded, x ∈ K; whilst the conditions for recurrence will be met if there exists a compact K with $\text{E}\text{{6X}{}_{\mathrm{n+1}}\text{6 ? 6X}{}_{\mathrm{n}}\text{6 ∣ X}{}_{\mathrm{n}}\text{= x} ? 0}$ for all x outside K. An application to queueing theory is given.     

γ-sets and other singular sets of real numbers

A family of $\text{J}$ of open subsets of the real line is called an ω-cover of a set X iff every finite subset of X is contained in an element of $\text{J}$. A set of reals X is a γ-set iff for every ω-cover $\text{J}$ of X there exists $\text{〈D}{}_{\mathrm{n}}\text{: n < ω〉?}\text{J}{}^{\mathrm{\omega }}$ such that

In this paper we show that assuming Martin's axiom there is a γ-set X of cardinality the continuum.     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

L-spaces and S-spaces in P(ω)

We show that CH implies that $\text{P}\text{(ω)}$, when equipped with the Vietoris topology, has a subspace which is an L-space and a subspace which is an S-space. This is an immediate consequence of the following purely combinatorial result: CH implies the existence of an ω₁-sequence 〈x_α: α < ω₁〉 in $\text{P}\text{(ω)}$ such that (1) if α<β<ω₁, then $\text{X}{}_{\mathrm{\beta }}\text{?}{}^1\text{X}{}_{\mathrm{\alpha }}$; (2) if I ?ω₁ is unaccountable, then there are distinct α, β ∈ I with X_β ?X_α.     

Collections of paths and rays in the plane which fix its topology

A collection $\text{F}$ of proper maps into a locally compact Hausdorff space X fixes the topology of X if the only locally compact Hausdorff topology on X which makes each element of $\text{F}$ continuous and proper is the given topology. In I²=[-1, 1]×[-1, 1], neither the collection of analytic paths nor the collection of regular twice differentiable paths fixes the topology. However, in I², both the collection of C^∞ arcs and the collection of regular C¹ arcs fix the topology. In $\text{I}{}^2\text{=[?1,1]×[?1,1], the collection of polynomial rays together with any collection of paths does not fix the topology. However, in}\text{R}{}^2$, the collection of regular injective entire rays together with either the collection of C^∞ arcs or the collection of regular C¹ arcs fixes the topology.     

Compactness and collective compactness in spaces of compact operators

This is a study of compactness in (a) spaces K_b(X, Y) of compact linear operators, (b) injective tensor products $\text{X}\text{}\text{?}\text{bo}{}_{\mathrm{?}}\text{Y}$, and (c) spaces L_c(X, Y) of continuous linear operators, and its various relationships with equicontinuity and collective compactness. Among the applications is a result on factoring compact sets of compact operators compactly and uniformly through one and the same reflexive Banach space.     

Indestructibility of compact spaces

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to _ω1${\omega }_1$-sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba?s “_ℵ1${\aleph }_1$-Borel Conjecture” is equiconsistent with the existence of an inaccessible cardinal.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

Domination by metric spaces

Following the definition of domination of a topological space X by a metric space M introduced by Cascales, Orihuela and Tkachuk (2011) in [3], we define a topological cardinal invariant called the metric domination index of a topological space X   as minimum of the set {w(M):M is a metric space that dominates X}$\{w(M):M\text{is a metric space that dominates}X\}$. This invariant quantifies or measures the concept of M-domination of Cascales et al. (2011) [3]. We prove (in ZFC) that if K   is a compact space such that _Cp(K)$C_p(K)$ is strongly dominated by a second countable space then K is countable. This answers a question by the authors of Cascales et al. (2011) [3].     

On c-realcompact spaces

Abstract

The c-realcompact spaces are fully studied and most of the important and well-known properties of realcompact spaces are extended to these spaces. For a zero-dimensional space X, the space υ₀X, which is the counterpart of υX, the Hewitt realcompactification of X, is introduced and studied. It is shown that υ₀X, which is the smallest c-realcompact space between X and β₀X, plays the same role (with respect to C_c(X)) as υX does in the context of C(X). It is proved for strongly zero-dimensional spaces, c-realcompact spaces, realcompact spaces and N-compact spaces coincide. In particular, if X is a strongly zero-dimensional space, then υX = υ₀X. It is obsesrved that a zero-dimensional space X is pseudocompact if and only if C_c(X) = C*_c(X), or equivalently if and only if υ₀X = β₀ X. In particular, a zero-dimensional pseudocompact space is compact if and only if it is c-realcompact. It is shown that Lindelöf spaces, subspaces of the one-point compactification (resp., Lindelöffication) of a discrete space with a nonmeasurable cardinal, are c-realcompact space. If X is a pseudocompact space, it is observed that C(X) = C_c(X) if and only if βX is scattered. Finally, the simplest possible proof (with reasoning) among the known proofs, of the well-known fact that discrete spaces of cardinality less than or equal to that of the continuum are realcompact, is given.     

Every scattered space is subcompact

We prove that every scattered space is hereditarily subcompact and any finite union of subcompact spaces is subcompact. It is a long-standing open problem whether every ?ech-complete space is subcompact. Moreover, it is not even known whether the complement of every countable subset of a compact space is subcompact. We prove that this is the case for linearly ordered compact spaces as well as for ω  -monolithic compact spaces. We also establish a general result for Tychonoff products of discrete spaces which implies that dense _Gδ$G_{\delta }$-subsets of Cantor cubes are subcompact.     

Stability and locally exact differentials on a curve

We show that the locally free sheaf $\text{B}{}_1\text{?F}{}_1\text{(Ω}{}^1{}_{\mathrm{X}}\text{)}$ of locally exact differentials on a smooth projective curve of genus g?2 over an algebraically closed field k of characteristic p is a stable bundle. This answers a question of Raynaud. To cite this article: K. Joshi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Remote points in large products

A point p ∈ βX\X is a remote point of X if p? cl_βXD for any nowhere dense D ? X. Van Douwen, and independently Chae and Smith, have shown that each non-pseudocompact space of countable π-weight has a remote point. Van Mill showed that many spaces of π-weight ω₁, such as ω×2^ω₁ also have remote points.We show that arbitrarily large products of spaces with countable π-weight which are not pseudocompact have remote points. In particular, ω×2^? for any infinite cardinal ?.